So, talk about DirectX and, specifically, DirectX 11 features for gaming (such as tesselation) are common on these forums and I don't really know what it is. I assumed it must have been the mark of higher-end gaming PCs and I looked closely at the DXDiag (Does this "DX" in this stand for "DirectX") of this laptop that can't even run Oblivion... and found that its supported version of DirectX is DX11... so that throws my rough assumption out of the window.
So, what is DirectX?
A multimedia API proprietary to Windows and Microsoft
How does it benefit non-gaming related PC functions (In other words, what use would this laptop have for DX11)?
http://en.wikipedia.org/wiki/DirectX_Video_Acceleration -- Also, Aero required DX. Many applications will make calls to DX since you are pretty much guaranteed to have it, so they don't have to worry about any dependency issues, so it can be a useful shortcut for your application if there is a relevant library in DX.
How does DirectX differ from OpenGL?
DirectX is a full multimedia API, whereas OpenGL is just a graphics API (as the name implies). In order to reach the functionality of DirectX, you must use other libraries. There is a whole Open*L family (though very few are actually really related to each other, but rather follow similar design models, but were designed by different people) and other projects that aim for this, such as http://en.wikipedia.org/wiki/Simple_DirectMedia_Layer. Direct3D is the equivocal DX library to OpenGL.
In addition, what is the difference between 32-bit and 64-bit... anything in relation to computers? Only the 64-bit version of internet explorer 9 seems to work with this computer and it seems weaker than the 32-bit version, even though I would think it would be better. For example, videos from sites outside of Youtube aren't watchable, for some reason. Why is that?
The processor architecture is different, granted, the difference between x86 and x86-64 isn't huge (x86-64 is an extension of x86 rather than a completely new architecture like IA64). A 64-bit processor is any processor with a 64-bit register and supports 64-bit floating numbers.
A 64-bit application is any application that is compiled against a 64-bit architecture.
The reason Internet Explorer 64-bit is much worse is that the 32-bit version has a highly optimized javascript engine, but it wasn't rewritten and compiled for the 64-bit version, so the 64-bit version shipped with a different javascript engine that is significantly slower. Also many browser plugins don't support 64-bit browsers yet, as such much web content doesn't work in them. Those other sites probably aren't compatible with the 64-bit version of flash, which isn't even at a stable release yet.
For one last question, is there any normal reason as to a why computer should be using 20-50% of its CPU and 50% of its physical memory while seemingly doing nothing but showing the task manager? That's been happening with this computer for as long as I can remember using it, now, and I'm just curious to know if it's normal.
It definitely shouldn't be idling with that much CPU usage, sounds like you have some zombie process svcking up CPU cycles, I'd open task manager and see what process it is, research it, and kill it if it isn't something important.
Memory is a different matter. Modern OSes keep processes and recently used data in memory instead of flushing it out when you are done. This allows you to relaunch applications and reopen files significantly faster. There is nothing wrong with 50% memory usage, as if any one application needs more, it'll just empty memory currently allocated to applications you aren't using and give that memory to the ones you are.